Topological description of the Borel probability space

TL;DR

This paper investigates various topologies on the space of Borel probability measures, establishing conditions for their equivalence, separability, and metrizability, and providing criteria for relative compactness in compact metric spaces.

Contribution

It offers new insights into the relationships between different topologies on Borel probability spaces and characterizes conditions for their topological properties.

Findings

Vague topologies are equivalent on LCH spaces.

Setwise topologies are equivalent regardless of ambient space topology.

Conditions for separability and metrizability are provided.

Abstract

We study properties of some popular topology on the space of Borel probabilities on a topological ambient space in this paper. We show that the two types of popular vague topology are equivalent to each other in case the ambient space is LCH. The two types of setwise topology induced from two equivalent descriptions of setwisely sequential convergence of probability measures are also equivalent to each other regardless of the topology on the ambient space. We give explicit conditions for the two types of vague topology and the two types of setwise topology to be separable or metrizable on the space of Borel probabilities. These conditions are either in terms of the cardinality of the elementary events in the Borel $\sigma$-algebra or some direct topological assumptions on the ambient space. We give an necessary and sufficient condition for families of probability measures to be setwisely relatively compact in case the ambient space is a compact metric space. There are some extending problems and heuristic schemes on formulating new topologies on the space of Borel probabilities at the end of the work.